U.S. patent number 4,503,435 [Application Number 06/505,100] was granted by the patent office on 1985-03-05 for multibeam antenna arrangement with minimal astigmatism and coma.
This patent grant is currently assigned to AT&T Bell Laboratories. Invention is credited to Corrado Dragone.
United States Patent |
4,503,435 |
Dragone |
March 5, 1985 |
**Please see images for:
( Certificate of Correction ) ** |
Multibeam antenna arrangement with minimal astigmatism and coma
Abstract
The present invention relates to a multibeam antenna arrangement
having minimal aberration of astigmatism and coma over a wide area
of the focal surface of the antenna. The present antenna comprises
a plurality of N reflectors arranged confocally in a sequence along
a feed axis of the antenna and at least one feed disposed in the
vicinity of a focal point on the focal surface. The reflectors and
the at least one feed are further arranged to provide an equivalent
centered antenna arrangement with the longitudinal axis of the feed
corresponding to an equivalent axis of the centered arrangement for
eliminating astigmatism. Coma is then eliminated by deforming two
of the N reflectors in a predetermined manner.
Inventors: |
Dragone; Corrado (Little
Silver, NJ) |
Assignee: |
AT&T Bell Laboratories
(Murray Hill, NJ)
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Family
ID: |
26997521 |
Appl.
No.: |
06/505,100 |
Filed: |
June 16, 1983 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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352389 |
Feb 25, 1982 |
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Current U.S.
Class: |
343/781P;
343/914 |
Current CPC
Class: |
H01Q
19/191 (20130101) |
Current International
Class: |
H01Q
19/19 (20060101); H01Q 19/10 (20060101); H01Q
019/19 () |
Field of
Search: |
;343/779,781P,781CA,837,840,914 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Proc. of IEEE, vol. 59, No. 2, Feb. 1971, "A Reflector Antenna
Corrected for Spherical, Coma and Chromatic Aberrations", by A. R.
Panicali et al., pp. 311-312. .
AP-S Intnat'l. Symposium, vol. II, Seattle, Washington, 1979,
"Astigmatic Correction by a Deformable Subreflector", by W-X. Wong
et al., pp. 706-709. .
IEEE Trans. Antennas & Propagation, vol. AP-30, No. 3, May
1982, "A First Order Treatment of Aberrations in Cassegrainian and
Gregorian Antennas", by C. Dragone, pp. 331-339..
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Primary Examiner: Lieberman; Eli
Attorney, Agent or Firm: Pfeifle; Erwin W.
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATION
This application is a continuation-in-part of application Ser. No.
352,389 filed Feb. 25, 1982, abandoned.
Claims
What is claimed is:
1. A multibeam antenna arrangement with minimal aberrations due to
astigmatism and coma, comprising: a plurality of N sequentially
confocal reflectors including N+1 separate focal points comprising
at least a curved focusing offset main reflector capable of
bidirectionally reflecting a beam of radiated electromagnetic
energy between the N.sup.th and the N+1 focal points along a feed
axis thereof, and a subreflector disposed along the feed axis of
the main reflector comprising a curved reflecting surface capable
of bidirectionally reflecting said beam between said N.sup.th and
an N-1 points of the N+1 separate focal points; and
at least one feedhorn disposed at or in the vicinity of a first
focal point of said N+1 focal points and oriented with a
longitudinal axis thereof coincident with an equivalent axis of the
plurality of N sequential confocal reflectors, the equivlent axis
being an axis of revolution which passes through the first focal
point of an equivalent reflecting surface which is capable of
producing after a single reflection the same field distribution
over the reflected wavefront as that of the plurality of N
sequential confocal reflectors
characterized in that
the reflecting surface of each of two of the plurality of N
sequential confocal reflectors are deformed with a separate
deformation coefficient, C.sub.n specifying the displacement,
Z.sub.n, of each associated reflecting surface in the Z.sup.th
direction according to the relationship
where P.sub.n is the distance from the Z axis at a focal point of
the reflector, and the deformation coefficient for each of the two
reflecting surfaces is specified by the relationship ##EQU6## where
the magnification of any reflector ##EQU7## with l.sub.2n-1 and
l.sub.2n-2 being the distances along the feed axis of the antenna
arrangement to a central point on the reflector from the focal
point of the reflector nearest and furthest, respectively, from the
first focal point of the antenna arrangement; n.sub.1 and n.sub.2
designate the number of the nearest and furthest reflector,
respectively, along the feed axis from the first focal point of the
antenna arrangement to be deformed; i is the angle of incidence of
a ray propagating between the two focal points of a reflector which
impinges the central point of the reflector n;
(M.sub.n.sbsb.1.sub.+1 . . . M.sub.n.sbsb.2) represents the product
of the magnifications of the reflectors n.sub.1 +1 to and including
reflector n.sub.2 ; M is the total magnification of the antenna
arrangement; and (M.sub.1 . . . M.sub.n.sbsb.2) is the product of
the magnifications of the first reflector up to and including the
n.sub.2 reflector of the antenna arrangement.
2. A multibeam antenna arrangement according to claim 1 wherein N=2
and each of the two reflectors is deformed in accordance with its
predetermined deformation coefficient.
3. A multibeam antenna arrangement according to claim 2 wherein the
two reflectors comprise a deformed parabolic main reflector and a
deformed subreflector and the magnification, M.sub.1, of the
subreflector is defined by ##EQU8## where i.sub.1 is the angle of
incidence of a ray impinging a central point on the reflecting
surface of the subreflector, i.sub.2 is the angle of incidence of
the ray impinging a central point on the reflecting surface on the
main reflector, d is the distance between the central points on the
reflecting surfaces of the main reflector and subreflector, and
l.sub.2 is the focal length of the main reflector.
Description
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to aplanatic reflector arrangements
for offset multibeam ground station or satellite antennas and, more
particularly, to multibeam antenna arrangements comprising N
reflectors disposed in a particular sequential arrangement with two
of the reflectors being slightly deformed in a predetermined manner
to cause substantial elimination of the aberrations of astigmatism
and coma over a wide area of the focal surface of the antenna.
2. Description of the Prior Art
Of considerable interest in practice is the problem of modifying an
existing or a new antenna design so as to reduce or substantially
eliminate aberrations which might be produced. More particularly,
Cassegrainian and Gregorian reflector arrangements are needed for
multibeam ground station and satellite antennas. In these antennas,
an arrangement of two reflectors, a paraboloid and either a
hyperboloid or an ellipsoid, is combined with several feeds
disposed in the vicinity of a focal point. Each feed produces a
beam whose direction is determined by the feed displacement from
the focal point. This displacement has been found to normally cause
aberrations due primarily to astigmatism and coma. Various
arrangements have been derived to correct one or more of such
aberrations in antennas. One such arrangement was for Spherical,
Coma and Chromatic Aberrations" by A. R. Panicali et al in
Proceedings of the IEEE, Vol. 59, No. 1, February, 1971, at pp.
311-312 where a corrugated reflector with varying depths of
corrugations was suggested.
In the article "Astigmatic Correction by a Deformable Subreflector"
by W-Y Wong et al in AP-S International Symposium, Vol. II,
Seattle, Wash., 1979, at pp. 706-709, a mechanically deformable
subreflector is suggested for providing a first order astigmatic
correction. Other astigmatic correction arrangements have been
disclosed in, for example, U.S. Pat. No. 4,145,695 issued to M. J.
Gans on Mar. 20, 1979 and U.S. Pat. No. 4,224,626 issued to R. L.
Sternberg on Sept 23, 1980. The Gans patent provides an astigmatic
launcher reflector for each off-axis feedhorn which has a reflector
having a curvature and orientation of its two orthogonal principal
planes of curvature which are chosen in accordance with specific
relationships. The Sternberg patent uses a lens having an
elliptical periphery and surfaces defined by a system of nonlinear
partial differential equations.
U.S. Pat. No. 4,166,276 issued to C. Dragone on Aug. 28, 1979
relates to an offset antenna having improved symmetry in the
radiation pattern and comprising a curved focusing main reflector,
at least two conic subreflectors and a feedhorn, the combination of
these elements being oriented such that the feedhorn is disposed at
the focal point of the combined confocal reflectors and in a manner
to coincide with the equivalent axis of the antenna system. Such
arrangement eliminates astigmatism to a first order
approximation.
More recently, U.S. patent appln. Ser. No. 209,943 filed on Nov.
24, 1980 for T. Chu, now U.S. Pat. No. 4,339,757, and U.S. patent
appln. Ser. No. 209,944 filed on Nov. 24, 1980 for E. A Ohm, now
U.S. Pat. No. 4,343,004, where each disclose different astigmatic
correction means comprising a first and a second doubly curved
subreflector which are curved in orthogonal planes to permit the
launching of an astigmatic beam of constant size and shape over a
broadband range.
The foregoing astigmatic correction arrangements, however, are
primarily designed to provide such correction in a very limited
portion of the focal surface. The problem remaining in the prior
art is to provide an antenna arrangement for multibeam transmission
which will correct for astigmatism and also coma over a wide area
of the focal surface of the antenna arrangement.
SUMMARY OF THE INVENTION
The foregoing problem has been solved in accordance with the
present invention which relates to aplanatic reflector arrangements
for offset multibeam ground station or satellite antennas and, more
particularly, to multibeam antenna arrangements comprising N
reflectors disposed in a predetermined manner to cause substantial
elimination of the aberrations of astigmatism and coma over a wide
area of the focal surface of the antenna.
It is an aspect of the present invention to provide an offset
antenna with astigmatism and coma free operation in the general
area of a focal point and at substantially reduced values beyond
such area by confocally arranging a plurality of N reflectors in
sequence to provide an equivalent essentially centered antenna
arrangement which is free of astigmatism, and doubly curving two of
the reflectors in a predetermined manner to also eliminate
coma.
Other and further aspects of the present invention will become
apparent during the course of the following description and by
reference to the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
Referring now to the drawings, in which like numerals represent
like parts in the several views:
FIG. 1 is a typical prior art antenna system where a spherical wave
from a focal point F.sub.0 is transformed into a plane wave by
three confocal reflectors;
FIG. 2 is a diagram of a method of determining the equivalent axis
of a reflector via a reflected ray emanating from a foci of the
reflector;
FIG. 3 illustrates the method of FIG. 2 extended to determine the
equivalent axis of a confocal sequence of N reflectors;
FIG. 4 illustrates a simple method for determining the equivalent
axis of a sequence of N confocal reflectors where the last
reflector, .SIGMA..sub.N, is a paraboloid in FIG. 3;
FIG. 5 is an exemplary illustrative antenna reflector arrangement
including four ellipsoid reflectors for transforming an input ray
into an output ray in accordance with the present invention;
FIG. 6 illustrates any one of the ellipsoids of FIG. 5 which is to
be deformed as specified in the present invention; and
FIG. 7 illustrates a compact reflector arrangement in accordance
with the present invention.
DETAILED DESCRIPTION
In accordance with the present invention, a multibeam antenna
arrangement is provided which substantially eliminates the
aberrations of astigmatism and coma. In the present arrangement,
astigmatism is substantially eliminated for feeds in the vicinity
of a focal point by centering the antenna aperture with respect to
an equivalent paraboloid axis. Having achieved an effectively
centered arrangement, coma is then substantially eliminated by
doubly curving two of the reflecting surfaces of the antenna
arrangement in a predetermined manner as will be explained
hereinafter.
A preferred technique for achieving an effectively centered antenna
arrangement in an offset antenna is described in U.S. Pat. No.
4,166,276 issued to C. Dragone on Aug. 28, 1979 and briefly
discussed hereinbefore. In accordance with the patented
arrangement, perfect performance in cross-polarization
discrimination and elimination of astigmatism to a first order
approximation is achieved in an antenna system by disposing a
symmetrical feedhorn at the focal point of the antenna system such
that the longitudinal axis of the feedhorn coincides with the
equivalent axis of the antenna system. The description which
follows is intended to provide the necessary background and
explanation for the various arrangements of antenna elements to
achieve a centered arrangement with astigmatism free operation in
the far field of the antenna and is a condensed explanation of the
patented Dragone arrangement.
In FIG. 1 a typical antenna system is shown comprising a feedhorn
10 disposed at a focal point F.sub.0 of the antenna system and
three reflectors designated .SIGMA..sub.1 and .SIGMA..sub.3 to
produce a spherical wave after each reflection which passes through
focal points F.sub.1 and F.sub.3, respectively. Thus, in general,
if F.sub.N is the focal point after the N.sup.th reflection, the
N.sup.th reflector .SIGMA..sub.N transforms a spherical wave
centered at the focal point F.sub.N-1, into a spherical wave
centered at focal point F.sub.N. It is to be understood that any of
the focal points F.sub.0 to F.sub.N may be at .infin., in which
case the corresponding spherical waves become plane waves. This
condition is shown in FIG. 1 by placing F.sub.3 at .infin. which
requires reflector .SIGMA..sub.3 to be a paraboloid.
It can be demonstrated that a sequence of confocal reflectors as
shown, for example, in FIG. 1 always has an equivalent single
reflector which will be either an ellipsoid, hyperboloid or
paraboloid. This equivalent reflector produces, after a single
reflection the same reflected wave pattern as was produced by the
given sequence of reflectors. This means that the field
distribution over a wavefront reflected by the equivalent single
reflector will coincide with the field distribution over the
corresponding wavefront produced by the given sequence of
reflectors. It is to be understood that such equivalent single
reflector does not of necessity coincide with the location of any
one of the given sequence of reflectors or that the direction of
the wavefront produces by the single equivalent reflector has to
correspond to the direction of the wavefront produced by the given
sequence of reflectors. The only correlation between the single
equivalent reflector and the given sequence of reflectors is that
the field distribution over the wavefront produced by each of the
arrangements are the same.
In accordance with the foregoing explanations, for purposes of
determining the properties of the reflected wave, it is possible to
replace the N confocal reflectors of FIG. 1 with an equivalent
reflector (not shown). The equivalent reflector has an axis of
revolution which passes through focal point F.sub.0 and will
hereinafter be referred to as the "equivalent axis". The equivalent
axis for the three reflectors of FIG. 1 may, for example, be in the
direction shown in FIG. 1. How the equivalent axis is determined
will be more clearly shown hereinafter. It is to be understood that
in order for the symmetry of the incident beam to be preserved, the
principal ray must coincide with the equivalent axis, where the
principal ray is that ray which corresponds to the longitudinal
axis of the feedhorn disposed at focal point F.sub.0. Since, in
theory, it is possible to travel along the equivalent axis in two
opposite directions, two opposite orientations can be chosen for
the principal ray. Suffice it to say, that for symmetry to be
preserved, and in turn to eliminate cross-polarization components
in the wavefront reflected by reflector .SIGMA..sub.3 in FIG. 1,
feedhorn 10 should be reoriented to have its longitudinal axis
coincide with the equivalent axis.
For a clear understanding of the definition and derivation of the
equivalent axis, the single reflector .SIGMA..sub.1 as shown in
FIG. 2 will be considered. If the reflector .SIGMA..sub.1 and one
of its foci F.sub.0 are known, but the exact location of the axis
of .SIGMA..sub.1 is not known and must be found, then the following
procedure may be used. A ray emanating from foci F.sub.0 is
reflected twice by .SIGMA..sub.1 as shown in FIG. 2 where the
construction of the complete reflector .SIGMA..sub.1 is also shown.
Where s and s" are the initial and final direction of the ray,
respectively, after two reflections by .SIGMA..sub.1, then it can
be seen that s will only equal S" when the ray coincides with the
axis of the reflector. Therefore, by searching for a ray which
satisfies this condition, the axis of the reflector can be found.
As can also be seen from FIG. 2, two such rays can satisfy the
condition where s=s", the one shown in the Figure and the one which
emanates from F.sub.0 in a direction opposite to that shown in FIG.
2 for the axial ray.
The previous description can also be extended to determine the
equivalent axis for a confocal sequence of reflectors .SIGMA..sub.1
to .SIGMA..sub.N as shown in FIGS. 3 and 4 where N=3. This is
possible since, as was stated previously, a confocal sequence of
reflectors has an equivalent single reflector. Thus, to determine
the equivalent axis of a confocal sequence of reflectors, a ray
from focal point F.sub.0 with a direction s must be reflected twice
by each of the reflectors .SIGMA..sub.1 to .SIGMA..sub.N such that
s=s". The two reflections at each reflector indicates a total of 2N
reflections in the original configuration and the first N
reflections occur in the order .SIGMA..sub.1, . . . , .SIGMA..sub.N
while the last N reflections have the reverse order. The final ray
has a direction s" which is the same direction s as the original
ray when the original ray was launched coincident with the
equivalent axis of the confocal sequence of reflectors.
As shown in FIG. 3 s=s" and, therefore, the ray through focal point
F.sub.0 gives the correct orientation of the equivalent axis and,
in turn, the direction of the principal ray for which symmetry is
preserved. More particularly, the path of the ray in FIG. 3 is
closed after 2N refelctions and will retrace the original path
during each subsequent 2N reflections. This closed path, which
determines the equivalent axis, will hereinafter be referred to as
the "cental path" and the two rays which proceed along the central
path in opposite senses will be referred to as "central rays".
The condition that s=s" leads to a straightforward geometrical
procedure for determining the equivalent axis when the
.SIGMA..sub.N reflector is a paraboloid as shown in FIG. 4. In FIG.
4 it is shown that when the last reflector .SIGMA..sub.N is
replaced by a concave paraboloid reflector in, for example, FIG. 3,
the final ray direction after two reflections therefrom becomes
independent of the initial direction towards the first reflection
therefron. Therefore, the final ray after the second reflection
coincides with the paraboloid axis and has a direction going from
focus F.sub.N-1 towards the vertex V of the paraboloid
.SIGMA..sub.N.
Having substantially eliminated astigmatism with an equivalent
centered antenna reflector arrangement, any phase error produced
over the antenna aperture is a function of the aperture coordinate
x,y and is due to coma aberration. This residual aberration can be
reduced by increasing the equivalent focal length f. However, this
would increase the feed dimensions, and also the separation between
feeds corresponding to different beams in a multibeam antenna. In
accordance with the present invention, coma is substantially
eliminated without increasing the focal length, f, by slightly
deforming two of the antenna reflectors. For the special case of a
two reflector Cassegrainian or Gregorian antenna, both reflectors
can be modified. Alternatively, the subreflector can be replaced
with two deformed reflectors without modifying the main reflector,
or the two reflector antennas can be combined with two additional
deformed reflectors. Additionally, it is to be understood that two
sequential reflectors need not be deformed, although permissible,
but that any two of the N reflectors are deformed as outlined
hereinafter no matter where in the sequence along the feed
axis.
For a clear understanding of the necessary deformations of two of a
sequence of N reflectors to overcome coma once a centered
arrangement is achieved, an exemplary sequence of four reflectors,
where N=4, will now be considered to primarily define terms used
hereinafter in accordance with the present invention. In FIG. 5,
reflectors 1-4 are arranged confocally where reflector 1 has a
first focal point F.sub.0 on the focal surface where, for example,
feedhorn 10 of FIG. 1 would be disposed along the equivalent axis.
A principal ray 50 emanating from first focal point F.sub.0 is
reflected at a central point I.sub.1 on reflector 1 with an angle
of incidence 2i.sub.1 and passes through second focal point F.sub.1
of reflector 1. Focal point F.sub.1 is also a focal point of
reflector 2 and the principal ray 50 is reflected at a central
point I.sub.2 of reflector 2 with an angle of incidence 2i.sub.2
and passes through a second focal point F.sub.2 of reflector 2,
which second focal point F.sub.2 is also a first focal point of
reflector 3. The principal ray is similarly reflected by reflectors
3 and 4 and passes through the second focal point F.sub.4 of
reflector 4 which is the F.sub.N+1 focal point of the arrangement.
The length F.sub.0 to I.sub.1 is designated l.sub.0, the length
I.sub.1 to F.sub.1 is designated l.sub.1, the length F.sub.1 to
I.sub.2 is designated l.sub.2 and so forth with the length I.sub.4
to F.sub.4 being designated l.sub.7.
An optical system satisfying Abbe's sine condition as described in
greater detail in the book Principles of Optics, by M. Born and E.
Wolf, Pemagon, N.Y., 1959 in Section 4.10 at pages 197-200, is
called aplanatic and is free of aberrations. In accordance with the
present invention, two of N reflectors are slightly deformed as
will be described to substantially eliminate coma and provide an
aplanatic arrangement. To achieve such aplanatic antenna
arrangement using the exemplary arrangement of FIG. 5, it is to be
understood that any two of such reflectors 1-4 can be deformed.
FIG. 6 is used to define the deformation necessary for any of the
two reflectors.
In FIG. 6, an n.sup.th reflector, representing any one of the two
reflectors to be deformed, is shown having a first focal point
F.sub.n-1, a second focal point F.sub.n, a central point on the
reflector I.sub.n, an angle of incidence 2i.sub.n where the length
F.sub.n-1 to I.sub.n is designated l.sub.2n-2 and the length
I.sub.n to F.sub.n is designated l.sub.2n-1. If, for example,
reflector 2 of FIG. 5 were to be deformed, then n=2 and in FIG. 6
I.sub.n =I.sub.2, F.sub.n-1 =F.sub.1, F.sub.n =F.sub.2, l.sub.2n-2
=l.sub.2 and l.sub.2n-1 =l.sub.3, which corresponds to the elements
associated with reflector 2 in FIG. 5. It is to be understood that
in FIG. 6 both lengths are positive in value since both foci are
disposed in front of the n.sup.th reflector. However, if one of the
two foci is behind the reflector, then the corresponding length is
negative.
The magnification of the n.sup.th reflector, M.sub.n, is defined by
##EQU1## For purposes of illustration, it will be assumed that the
n.sup.th reflector is derived from an ellipsoid or hyperboloid
defined by the equation ##EQU2## where C.sub.n is the coefficient
of deformation of the n.sup.th reflector.
To determine the coefficient of deformation of a first and a second
reflector of the sequence of reflectors the designations n.sub.1
and n.sub.2 will be used to represent the first and second deformed
reflectors, respectively, hereinafter. Coma free operation in an
equivalent centered antenna arrangement is accomplished by
deforming n.sub.1 and n.sub.2 in accordance with the coefficients
of deformation derived from the equation: ##EQU3## where
(m.sub.n.sbsb.1.sub.+1 . . . M.sub.n.sbsb.2) represents the product
of the magnifications of the reflectors n.sub.1 +1 to and including
reflector n.sub.2 ; M is the total magnification of the antenna
arrangement; and (M.sub.1 . . . M.sub.n.sbsb.2) is the product of
the magnifications of the first reflector up to and including the
n.sub.2 reflector of the antenna arrangement.
Having thus substantially eliminated primary astigmatism and
primary coma, any residual phase error produced over the antenna
aperture by a feed placed in the vicinity of the focal point
F.sub.0 is very small since it can be shown to be proportional to
the square of the distance d.sub.f of the feed from the focus. In
the vicinity of the center of the aperture, the residual phase
error is due to astigmatism, with coefficient A.sub.2 being
proportional to d.sub.f.sup.2. In general, if the various
magnifications M.sub.n which specify the distances l.sub.2n-2 and
l.sub.2n-1 are chosen arbitrarily, a nonzero A.sub.2 will be
produced. It is, however, possible to choose, in general, the
magnifications M.sub.n so as to cause A.sub.2 to approximately
equal 0. This is important in those applications requiring large
displacements d.sub.f. Of greatest interest is the case of two
reflectors including a deformed subreflector and a deformed
parabolic main reflector arranged with a common plane of symmetry.
Then it is important to minimize A.sub.2 when the feed displacement
from the focus is orthogonal to the plane of symmetry since large
feed displacements are possible in this case without blocking the
aperture with the feed, or the subreflector whose dimensions must
be increased when large feed displacements must be accommodated.
For such an arrangement with d.sub.f orthogonal to the symmetry
plane, it can be shown that the residual coefficient A.sub.2 can be
made to vanish by choosing the subreflector magnification M.sub.1
so that ##EQU4## where d is the distance .vertline.I.sub.1 I.sub.2
.vertline. between the two reflectors shown in FIG. 7. It is to be
recalled that in order for primary astigmatism to be eliminated,
the angles of incidence i.sub.1 and i.sub.2 must satisfy the
condition ##EQU5## discussed in U.S. Pat. No. 4,166,276 issued to
C. Dragone on Aug. 28, 1979.
It is of interest in practice to satisfy condition (4) with a
compact arrangement in which the distance between the two
reflectors does not exceed appreciably the aperture diameter D.
This can be obtained, without violating condition (5), and without
blocking the aperture, only for certain values of the magnification
M.sub.1. More precisely, one finds that M.sub.1 must be closed to
0.5, in which case tani.sub.1 and tani.sub.2 have approximately the
same magnitude, but opposite sign. By choosing, for example,
M.sub.1 =0.5 and 2i.sub.1 .about.70 degrees, the arrangement of
FIG. 7 is obtained with 2i.sub.2 .about.-70 degrees, d=0.509
l.sub.1 and D=d. The equivalent focal length f is only
0.980.times.D, and therefore each feed has a relatively small
aperture, and the feed displacement from the focus is small, for a
given angular displacement .delta..theta. of the antenna beam from
the axis.
This arrangement, because it is virtually free of aberrations for
relatively large values of .delta..theta., is expected to be
particularly useful for ground stations with a wide field of view,
exceeding for instance .+-.10 degrees. The field of view obtainable
in FIG. 7 is expected to be comparable to that of the well-known
Schmidt camera.
* * * * *